(SCCCR) Geometry Overview
South Carolina College- and Career-Ready (SCCCR) Geometry Overview In South Carolina College- and Career-Ready (SCCCR) Geometry, students build on the conceptual knowledge and skills they mastered in previous mathematics courses in areas such as algebraic thinking, geometry, measurement, probability, and proportional reasoning. The Key Concepts in this course are listed below.
Constructions, Transformations, and Coordinate Geometry (G.CTC) Reasoning and Proof (G.RP) Lines and Angles (G.LA) Triangles (G.T) Quadrilaterals and Other Polygons (G.QP) Circles (G.C) Three-Dimensional Figures (G.TD)
Standards in the Key Concept Constructions, Transformations, and Coordinate Geometry are meant to be applied throughout the course in order for students to make critical connections among geometric relationships synthetically (without coordinates) and analytically (with coordinates). Students also construct logical arguments and formal proofs of geometric relationships throughout the course as they develop their deductive reasoning skills and understanding of more sophisticated theorems based on simpler axioms introduced early in the course. In this course students are expected to apply mathematics in meaningful ways to solve problems that arise in the workplace, society, and everyday life through the process of modeling. Mathematical modeling involves creating appropriate equations, graphs, diagrams, or other mathematical representations to analyze real-world situations and solve problems. Use of mathematical tools is important in creating and analyzing the mathematical representations used in the modeling process. In order to represent and solve problems, students should learn to use a variety of mathematical tools and technologies, such as a compass, a straightedge, graph paper, patty paper, graphing utilities, and dynamic geometry software.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 60
South Carolina College- and Career-Ready (SCCCR) Geometry
Constructions, Transformations, and Coordinate Geometry
Key Concepts
Standards The student will: G.CTC.1 Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Constructions should include but are not limited to: a. congruent segments and congruent angles; b. segment bisectors and angle bisectors; c. perpendicular lines and parallel lines; d. equilateral triangles; e. inscribed and circumscribed circles of a triangle; f. tangent lines from a point on a circle or to a circle from an exterior point. G.CTC.2 Understand and apply transformations. a. Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. b. Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations. c. Describe a sequence of transformations that maps a figure onto its image. d. Identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations. e. Demonstrate that two figures are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. f. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 61
G.CTC.3
Reasoning and Proof
G.CTC.4
Represent and analyze figures in the coordinate plane. a. Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. b. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. c. Derive the formulas for determining distance and midpoint and use those formulas to solve mathematical and real-world problems. d. Represent the sum and difference of two vectors geometrically using the parallelogram method. e. Use a scale drawing to determine the magnitude and direction of a resultant vector by direct measurement. f. Derive the standard equation of a circle given the center and radius using the definition of a circle and the distance formula. g. Determine the center and radius of a circle given the standard equation and write the standard equation of a circle given sufficient information for determining the center and radius. h. Rewrite the general form of the equation of a circle in standard form by completing the square. i. Graph circles on the coordinate plane and use circle properties to solve mathematical and real-world problems. Represent and analyze figures in a three-dimensional coordinate system. a. Graph points on a three-dimensional coordinate system and explain how each coordinate of the point indicates the distance from the origin along the corresponding axis. b. Determine the distance from a point to the origin in the three-dimensional coordinate system.
The student will: G.RP.1 Understand the axiomatic structure of geometry by using undefined terms, definitions, postulates, conjectures, theorems, and corollaries. G.RP.2 Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement. Determine the validity of a biconditional statement by analyzing the associated conditionals. G.RP.3 Distinguish between inductive and deductive reasoning. G.RP.4 Demonstrate that certain conjectures are false by producing counterexamples. G.RP.5
G.RP.6
Identify and explain, both symbolically and with examples, uses of the Law of Detachment and the Law of Syllogism and relate these laws of logic to the construction of a deductive proof. Construct logical arguments and proofs of theorems and other results in geometry, including coordinate proofs and proofs by contradiction. Express proofs in a form that justifies the reasoning, including two-column proofs, paragraph proofs, and flow charts.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 62
Lines and Angles Triangles
The student will: G.LA.1 Define angle, circle, perpendicular, parallel, and skew in terms of the undefined notions of point, line, plane, and distance, and use geometric figures to represent and describe real-world objects. G.LA.2 Prove and apply in mathematical and real-world contexts theorems about lines and angles, including but not limited to the following: a. vertical angles are congruent; b. when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and corresponding angles are congruent; c. when a transversal crosses parallel lines, consecutive interior angles are supplementary; d. any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; e. perpendicular lines form four right angles. G.LA.3 Apply properties of pairs of angles, including linear pairs, vertical angles, complementary angles, and supplementary angles, to solve problems and justify results. G.LA.4 Describe and identify the intersections of lines, planes, and other geometric figures. The student will: G.T.1 Understand and apply triangle congruency relationships. a. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-SideAngle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions. b. Prove theorems about the relationships within triangles, including the Midsegment Theorem, the Angle Sum Theorem, and Exterior Angle Theorem, and apply these relationships to solve problems. c. Verify experimentally the conclusions of the concurrency theorems for the medians, altitudes, angle bisectors, and perpendicular bisectors in triangles and apply these relationships to solve problems. d. Prove properties of equilateral and isosceles triangles and apply them to solve problems and justify results.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 63
G.T.2
Quadrilaterals and Other Polygons
G.T.3
Understand and apply triangle inequality relationships. a. Verify experimentally the conclusion of the Triangle Inequality Theorem using constructions or manipulatives and apply the theorem to solve problems and justify results. b. Prove the Hinge Theorem and its converse and apply them to solve problems and justify results. c. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. d. Prove the Triangle Proportionality Theorem, the Geometric Mean Theorem for right triangles, and the Angle Bisector Theorem and apply these theorems to solve problems and justify results. e. Use properties of similar triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of triangles. Understand and apply right triangle relationships. a. Prove the Pythagorean Theorem using triangle similarity and use the theorem and its converse to solve problems and justify results. b. Understand and apply properties of 45-45-90 and 30-60-90 triangles to solve problems and justify results. c. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. d. Determine the lengths of sides and the measures of angles of a right triangle by applying the trigonometric ratios sine, cosine, and tangent in real-world and mathematical problems using calculators, computers, or trigonometric tables. e. Explain and use the relationship between the sine and cosine of complementary angles. f. Derive the formula for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
The student will: G.QP.1 Understand and apply properties of special quadrilaterals. a. Prove theorems about parallelograms and apply those theorems to solve problems and justify results. Theorems include but are not limited to: parallelograms have congruent opposite angles; diagonals of a parallelogram bisect each other; rectangles have congruent diagonals; and rhombi have perpendicular diagonals. b. Prove that given quadrilaterals are parallelograms, rhombi, rectangles, squares, or trapezoids. Include coordinate proofs.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 64
Circles
G.QP.2
Understand and apply properties of polygons. a. Find measures of interior and exterior angles of polygons. Explain and justify the methods used. b. Apply congruence and similarity among quadrilaterals and other polygons to solve problems. c. Analyze how changes in the dimensions of quadrilaterals and regular polygons affect perimeter and area. d. Derive the formula for the area of any regular polygon in terms of its apothem and perimeter and understand how the limiting case of this formula leads to the formula for the area of a circle. Use these formulas to solve mathematical and real-world problems.
The student will: G.C.1 Understand and apply properties of circles and their parts. a. Define and identify the following terms and use the relationships among them to solve mathematical and real-world problems: radius, diameter, arc, measure of an arc, chord, secant, tangent, and concentric circles. b. Demonstrate that the radius of a circle is perpendicular to a tangent line at the point of tangency and use this relationship to solve mathematical and real-world problems. c. Identify and describe relationships among central angles, inscribed angles, circumscribed angles, and their intercepted arcs and use those relationships to solve mathematical and real-world problems. d. Prove that the pairs of opposite angles of a quadrilateral inscribed in a circle are supplementary and use this relationship to solve problems. e. Find the measure of line segments, angles, and intercepted arcs formed by the intersection of two secant lines, two tangent lines, or a secant line and a tangent line with a circle to solve mathematical and real-world problems. f. Demonstrate that all circles are similar. G.C.2 Understand and apply properties of circumferences and areas of circles. a. Solve mathematical and real-world problems involving the circumference and area of a circle and analyze how a change in radius affects circumference and area. b. Derive the formula for the length of an arc intercepted by a central angle and apply this relationship to solve mathematical and real-world problems. c. Derive the formula for the area of a sector and apply this relationship to solve mathematical and real-world problems. d. Use geometric probability to solve mathematical and real-world problems involving circles and polygons.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 65
Three-Dimensional Figures
The student will: G.TD.1 Understand and apply properties of three-dimensional solids. a. Use geometric shapes, their measures, and their properties to describe threedimensional solids and their symmetries. b. Draw a top-view, front-view, side-view, and an isometric view of a given threedimensional object. c. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. d. Describe the three-dimensional object generated by revolving a two-dimensional object about a line. G.TD.2 Understand and apply properties of surface-areas and volumes of three-dimensional solids. a. Derive surface area and volume formulas for prisms and cylinders and explain the relationship between these formulas and the surface area and volume formulas for pyramids and cones. b. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results, including problems that involve algebraic expressions, composite figures, and real-world applications. c. Apply geometric properties of solids, including prisms, pyramids, cylinders, cones, and spheres, to model and solve real-world problems. d. Analyze how changes in one or more dimensions affect the surface area and volume of a three-dimensional object.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 66
Constructions, Transformations, and Coordinate Geometry (G.CTC) Reasoning and Proof (G.RP) Lines and Angles (G.LA) Triangles (G.T) Quadrilaterals and Other Polygons (G.QP) Circles (G.C) Three-Dimensional Figures (G.TD)
Standards in the Key Concept Constructions, Transformations, and Coordinate Geometry are meant to be applied throughout the course in order for students to make critical connections among geometric relationships synthetically (without coordinates) and analytically (with coordinates). Students also construct logical arguments and formal proofs of geometric relationships throughout the course as they develop their deductive reasoning skills and understanding of more sophisticated theorems based on simpler axioms introduced early in the course. In this course students are expected to apply mathematics in meaningful ways to solve problems that arise in the workplace, society, and everyday life through the process of modeling. Mathematical modeling involves creating appropriate equations, graphs, diagrams, or other mathematical representations to analyze real-world situations and solve problems. Use of mathematical tools is important in creating and analyzing the mathematical representations used in the modeling process. In order to represent and solve problems, students should learn to use a variety of mathematical tools and technologies, such as a compass, a straightedge, graph paper, patty paper, graphing utilities, and dynamic geometry software.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 60
South Carolina College- and Career-Ready (SCCCR) Geometry
Constructions, Transformations, and Coordinate Geometry
Key Concepts
Standards The student will: G.CTC.1 Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. Constructions should include but are not limited to: a. congruent segments and congruent angles; b. segment bisectors and angle bisectors; c. perpendicular lines and parallel lines; d. equilateral triangles; e. inscribed and circumscribed circles of a triangle; f. tangent lines from a point on a circle or to a circle from an exterior point. G.CTC.2 Understand and apply transformations. a. Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. b. Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations. c. Describe a sequence of transformations that maps a figure onto its image. d. Identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations. e. Demonstrate that two figures are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. f. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 61
G.CTC.3
Reasoning and Proof
G.CTC.4
Represent and analyze figures in the coordinate plane. a. Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. b. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. c. Derive the formulas for determining distance and midpoint and use those formulas to solve mathematical and real-world problems. d. Represent the sum and difference of two vectors geometrically using the parallelogram method. e. Use a scale drawing to determine the magnitude and direction of a resultant vector by direct measurement. f. Derive the standard equation of a circle given the center and radius using the definition of a circle and the distance formula. g. Determine the center and radius of a circle given the standard equation and write the standard equation of a circle given sufficient information for determining the center and radius. h. Rewrite the general form of the equation of a circle in standard form by completing the square. i. Graph circles on the coordinate plane and use circle properties to solve mathematical and real-world problems. Represent and analyze figures in a three-dimensional coordinate system. a. Graph points on a three-dimensional coordinate system and explain how each coordinate of the point indicates the distance from the origin along the corresponding axis. b. Determine the distance from a point to the origin in the three-dimensional coordinate system.
The student will: G.RP.1 Understand the axiomatic structure of geometry by using undefined terms, definitions, postulates, conjectures, theorems, and corollaries. G.RP.2 Identify and determine the validity of the converse, inverse, and contrapositive of a conditional statement. Determine the validity of a biconditional statement by analyzing the associated conditionals. G.RP.3 Distinguish between inductive and deductive reasoning. G.RP.4 Demonstrate that certain conjectures are false by producing counterexamples. G.RP.5
G.RP.6
Identify and explain, both symbolically and with examples, uses of the Law of Detachment and the Law of Syllogism and relate these laws of logic to the construction of a deductive proof. Construct logical arguments and proofs of theorems and other results in geometry, including coordinate proofs and proofs by contradiction. Express proofs in a form that justifies the reasoning, including two-column proofs, paragraph proofs, and flow charts.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 62
Lines and Angles Triangles
The student will: G.LA.1 Define angle, circle, perpendicular, parallel, and skew in terms of the undefined notions of point, line, plane, and distance, and use geometric figures to represent and describe real-world objects. G.LA.2 Prove and apply in mathematical and real-world contexts theorems about lines and angles, including but not limited to the following: a. vertical angles are congruent; b. when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and corresponding angles are congruent; c. when a transversal crosses parallel lines, consecutive interior angles are supplementary; d. any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; e. perpendicular lines form four right angles. G.LA.3 Apply properties of pairs of angles, including linear pairs, vertical angles, complementary angles, and supplementary angles, to solve problems and justify results. G.LA.4 Describe and identify the intersections of lines, planes, and other geometric figures. The student will: G.T.1 Understand and apply triangle congruency relationships. a. Prove two triangles are congruent by applying the Side-Angle-Side, Angle-SideAngle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions. b. Prove theorems about the relationships within triangles, including the Midsegment Theorem, the Angle Sum Theorem, and Exterior Angle Theorem, and apply these relationships to solve problems. c. Verify experimentally the conclusions of the concurrency theorems for the medians, altitudes, angle bisectors, and perpendicular bisectors in triangles and apply these relationships to solve problems. d. Prove properties of equilateral and isosceles triangles and apply them to solve problems and justify results.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 63
G.T.2
Quadrilaterals and Other Polygons
G.T.3
Understand and apply triangle inequality relationships. a. Verify experimentally the conclusion of the Triangle Inequality Theorem using constructions or manipulatives and apply the theorem to solve problems and justify results. b. Prove the Hinge Theorem and its converse and apply them to solve problems and justify results. c. Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. d. Prove the Triangle Proportionality Theorem, the Geometric Mean Theorem for right triangles, and the Angle Bisector Theorem and apply these theorems to solve problems and justify results. e. Use properties of similar triangles to solve real-world and mathematical problems involving sides, perimeters, and areas of triangles. Understand and apply right triangle relationships. a. Prove the Pythagorean Theorem using triangle similarity and use the theorem and its converse to solve problems and justify results. b. Understand and apply properties of 45-45-90 and 30-60-90 triangles to solve problems and justify results. c. Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. d. Determine the lengths of sides and the measures of angles of a right triangle by applying the trigonometric ratios sine, cosine, and tangent in real-world and mathematical problems using calculators, computers, or trigonometric tables. e. Explain and use the relationship between the sine and cosine of complementary angles. f. Derive the formula for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
The student will: G.QP.1 Understand and apply properties of special quadrilaterals. a. Prove theorems about parallelograms and apply those theorems to solve problems and justify results. Theorems include but are not limited to: parallelograms have congruent opposite angles; diagonals of a parallelogram bisect each other; rectangles have congruent diagonals; and rhombi have perpendicular diagonals. b. Prove that given quadrilaterals are parallelograms, rhombi, rectangles, squares, or trapezoids. Include coordinate proofs.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 64
Circles
G.QP.2
Understand and apply properties of polygons. a. Find measures of interior and exterior angles of polygons. Explain and justify the methods used. b. Apply congruence and similarity among quadrilaterals and other polygons to solve problems. c. Analyze how changes in the dimensions of quadrilaterals and regular polygons affect perimeter and area. d. Derive the formula for the area of any regular polygon in terms of its apothem and perimeter and understand how the limiting case of this formula leads to the formula for the area of a circle. Use these formulas to solve mathematical and real-world problems.
The student will: G.C.1 Understand and apply properties of circles and their parts. a. Define and identify the following terms and use the relationships among them to solve mathematical and real-world problems: radius, diameter, arc, measure of an arc, chord, secant, tangent, and concentric circles. b. Demonstrate that the radius of a circle is perpendicular to a tangent line at the point of tangency and use this relationship to solve mathematical and real-world problems. c. Identify and describe relationships among central angles, inscribed angles, circumscribed angles, and their intercepted arcs and use those relationships to solve mathematical and real-world problems. d. Prove that the pairs of opposite angles of a quadrilateral inscribed in a circle are supplementary and use this relationship to solve problems. e. Find the measure of line segments, angles, and intercepted arcs formed by the intersection of two secant lines, two tangent lines, or a secant line and a tangent line with a circle to solve mathematical and real-world problems. f. Demonstrate that all circles are similar. G.C.2 Understand and apply properties of circumferences and areas of circles. a. Solve mathematical and real-world problems involving the circumference and area of a circle and analyze how a change in radius affects circumference and area. b. Derive the formula for the length of an arc intercepted by a central angle and apply this relationship to solve mathematical and real-world problems. c. Derive the formula for the area of a sector and apply this relationship to solve mathematical and real-world problems. d. Use geometric probability to solve mathematical and real-world problems involving circles and polygons.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 65
Three-Dimensional Figures
The student will: G.TD.1 Understand and apply properties of three-dimensional solids. a. Use geometric shapes, their measures, and their properties to describe threedimensional solids and their symmetries. b. Draw a top-view, front-view, side-view, and an isometric view of a given threedimensional object. c. Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. d. Describe the three-dimensional object generated by revolving a two-dimensional object about a line. G.TD.2 Understand and apply properties of surface-areas and volumes of three-dimensional solids. a. Derive surface area and volume formulas for prisms and cylinders and explain the relationship between these formulas and the surface area and volume formulas for pyramids and cones. b. Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results, including problems that involve algebraic expressions, composite figures, and real-world applications. c. Apply geometric properties of solids, including prisms, pyramids, cylinders, cones, and spheres, to model and solve real-world problems. d. Analyze how changes in one or more dimensions affect the surface area and volume of a three-dimensional object.
SCCCR-M PUBLIC REVIEW DRAFT 11/05/2014
Page 66
